Group #11772386 — subgroups

Subgroups — #11772386

Order 1152.

226 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63771	C₂	1	2	yes	maximal
2	C₂×C₁₆×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#3788	—	2	2	no	—
4	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	#63771	—	2	2	no	—
6	#63771	C₄	1	4	yes	—
7	C₂×C₃²×C₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
9	C₁₆×C₃⋊D₃	—	2	2	no	—
10	C₂×D₃×C₁₆	—	6	2	no	—
11	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
12	C₂×#622	—	4	2	no	—
13	#3788	—	2	4	no	—
14	#3788	—	4	2	no	—
15	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃⋊(C₃⋊C₈)	Q₈	1	1	yes	characteristic
17	C₈×C₃⋊D₃	C₂×C₄	1	2	yes	—
18	C₈×C₃⋊D₃	D₄	1	2	yes	—
19	C₃²×C₁₆	—	2	1	no	—
20	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
21	#1381	—	9	1	no	maximal, Sylow
22	C₂×D₃×C₈	—	6	2	no	—
23	C₂×C₃×C₁₆	—	2	2	no	—
24	C₂×C₃⋊C₁₆	—	6	2	no	—
25	D₃×C₁₆	—	6	8	no	—
26	C₂²×C₃⋊D₃	Q₁₆	1	1	yes	characteristic
27	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₃⋊Dic₃	D₈	1	1	yes	characteristic
29	C₄×C₃⋊D₃	C₂²⋊C₄	1	2	yes	—
30	C₄×C₃⋊D₃	SD₁₆	1	2	yes	—
31	#622	—	4	4	no	—
32	C₃²×C₈	C₄²	1	1	yes	characteristic
33	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
34	C₃⋊(C₃⋊C₈)	C₄⋊C₄	1	2	yes	—
35	C₂²×C₁₆	—	9	1	no	—
36	C₂×#108	—	9	2	no	—
37	C₂×C₄×D₃	—	6	2	no	—
38	C₂×C₃×C₈	—	2	2	no	—
39	C₂×C₃⋊C₈	—	6	2	no	—
40	D₃×C₈	—	6	4	no	—
41	D₃×C₈	—	6	4	no	—
42	C₃×C₁₆	—	2	4	no	—
43	C₃⋊C₁₆	—	6	4	no	—
44	C₂²×C₃²	#107	1	1	yes	characteristic
45	C₂×C₃⋊D₃	—	2	1	no	—
46	C₂×C₃⋊D₃	—	2	1	no	—
47	C₂×C₃⋊D₃	#103	1	2	yes	—
48	C₃²×C₄	#95	1	1	yes	characteristic
49	C₃²×C₄	#106	1	1	yes	characteristic
50	C₃⋊Dic₃	#102	1	2	yes	—
51	C₂×M₁₆	—	18	2	no	—
52	C₂²×C₈	—	9	1	no	—
53	C₂×C₁₆	—	9	1	no	—
54	C₂×C₁₆	—	18	2	no	—
55	C₂×C₁₆	#157	1	1	yes	characteristic
56	#108	—	9	4	no	—
57	#108	—	18	2	no	—
58	C₂²×D₃	—	6	2	no	—
59	C₂×C₃×C₄	—	2	2	no	—
60	C₂×Dic₃	—	6	2	no	—
61	C₄×D₃	—	6	4	no	—
62	C₄×D₃	—	6	4	no	—
63	C₃×C₈	—	2	2	no	—
64	C₃×C₈	—	2	2	no	—
65	C₃⋊C₈	—	6	4	no	—
66	C₂×C₃²	#340	1	1	yes	characteristic
67	C₂×C₃²	#368	1	1	yes	characteristic
68	C₂×C₃²	#368	1	1	yes	characteristic
69	C₃⋊D₃	—	2	2	no	—
70	C₂²×C₄	—	9	1	no	—
71	M₁₆	—	18	4	no	—
72	M₁₆	—	36	2	no	—
73	C₂×C₈	—	9	1	no	—
74	C₂×C₈	—	9	2	no	—
75	C₂×C₈	—	9	2	no	—
76	C₂×C₈	—	36	2	no	—
77	C₂×C₈	C₂×#157	1	1	yes	characteristic
78	C₁₆	—	2	1	no	—
79	C₁₆	—	18	1	no	—
80	C₂²×C₃	—	2	2	no	—
81	C₂×D₃	—	6	4	no	—
82	C₂×D₃	—	6	4	no	—
83	C₂×D₃	—	6	4	no	—
84	C₃×C₄	—	2	2	no	—
85	C₃×C₄	—	2	2	no	—
86	Dic₃	—	6	4	no	—
87	C₃²	#1381	1	1	yes	characteristic, complemented, Sylow
88	C₂³	—	9	1	no	—
89	C₂×C₄	—	9	1	no	—
90	C₂×C₄	—	9	2	no	—
91	C₂×C₄	—	9	2	no	—
92	C₂×C₄	C₄⋊#157	1	1	yes	characteristic
93	C₈	—	9	2	no	—
94	C₈	—	36	4	no	—
95	C₈	C₄×#157	1	1	yes	characteristic
96	C₈	C₂²⋊#157	1	1	yes	characteristic
97	C₂×C₃	—	2	2	no	—
98	C₂×C₃	—	2	2	no	—
99	C₂×C₃	—	2	2	no	—
100	D₃	—	6	8	no	—
101	C₂²	—	9	2	no	—
102	C₂²	—	18	1	no	—
103	C₂²	—	18	1	no	—
104	C₂²	#63769	1	1	yes	characteristic
105	C₄	—	9	2	no	—
106	C₄	#63768	1	1	yes	characteristic
107	C₄	#63781	1	1	yes	characteristic
108	C₃	—	2	2	no	—
109	C₂	—	18	2	no	—
110	C₂	#10590140	1	1	yes	characteristic
111	C₂	#10590141	1	1	yes	characteristic
112	C₂	#10590182	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#63771

C₂

yes

maximal

C₂×C₁₆×C₃⋊D₃

C₂

yes

maximal, characteristic

C₂×#3788

—

C₂×C₈×C₃⋊D₃

C₂²

yes

characteristic

#63771

—

#63771

C₄

yes

—

C₂×C₃²×C₁₆

C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

characteristic

C₁₆×C₃⋊D₃

—

C₂×D₃×C₁₆

—

C₂×C₄×C₃⋊D₃

D₄

yes

—

—

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

Q₈

yes

characteristic

C₈×C₃⋊D₃

C₂×C₄

yes

—

C₈×C₃⋊D₃

D₄

yes

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

#1381

—

maximal, Sylow

C₂×D₃×C₈

—

C₂×C₃×C₁₆

—

C₂×C₃⋊C₁₆

—

D₃×C₁₆

—

C₂²×C₃⋊D₃

Q₁₆

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₂×C₃⋊Dic₃

D₈

yes

characteristic

C₄×C₃⋊D₃

C₂²⋊C₄

yes

—

C₄×C₃⋊D₃

SD₁₆

yes

—

#622

—

C₃²×C₈

C₄²

yes

characteristic

C₃²×C₈

C₂²⋊C₄

yes

characteristic

C₃⋊(C₃⋊C₈)

C₄⋊C₄

yes

—

C₂²×C₁₆

—

C₂×#108

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃⋊C₈

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#107

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

#103

yes

—

C₃²×C₄

yes

C₃²×C₄

yes

C₃⋊Dic₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

—

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₃⋊D₃

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#157

yes

characteristic

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

C₃²

#1381

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊#157

yes

characteristic

C₈

—

C₈

—

C₈

C₄×#157

yes

characteristic

C₈

C₂²⋊#157

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

100

D₃

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

#63769

yes

characteristic

105

C₄

—

106

C₄

#63768

yes

characteristic

107

C₄

#63781

yes

characteristic

108

C₃

—

109

C₂

—

110

C₂

#10590140

yes

characteristic

111

C₂

#10590141

yes

characteristic

112

C₂

#10590182

yes

characteristic

	subgroup	G/H
center	C₂²	#63769
commutator	C₃²×C₈	C₄²
Frattini	C₂×C₈	C₂×#157
Fitting	C₂×C₃²×C₁₆	C₄
socle	C₂²×C₃²	#107
prime radical	C₂²×C₃⋊D₃	Q₁₆
cosocle	C₂×C₈×C₃⋊D₃	C₂²
hypercenter	C₂×C₁₆	#157
hypocenter	C₃²	#1381
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₂²

#63769

commutator

C₃²×C₈

C₄²

Frattini

C₂×C₈

C₂×#157

Fitting

C₂×C₃²×C₁₆

C₄

socle

C₂²×C₃²

#107

prime radical

C₂²×C₃⋊D₃

Q₁₆

cosocle

C₂×C₈×C₃⋊D₃

C₂²

C₂×C₁₆

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
87. C₃²	#1381	semidirect

H ≅ G/N

type

87. C₃²

#1381

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#63771	2
C₂×C₁₆×C₃⋊D₃	2
#1381	9

subgroup

index

C₂×#63771

C₂×C₁₆×C₃⋊D₃

#1381

Subgroups — #11772386

Conjugacy classes of subgroups

Complements

Maximal subgroups